International Journal of Advancements in Research & Technology, Volume 6, Issue 4, April-2017 ISSN 2278-7763 1 Set theory is the ultimate branch of Mathematics Mohammad Al- Amin Lecture (Mathematics) Department of Business Administration Z.H. Sikder University of Science & Technology [email protected] Abstract: Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to meet mathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The study investigates various types of set theory and their applications in solving real time problems. The main intent of the study is to explore different types of set theory and their dynamic usage in the different areas of problems. So the study highlights the dynamic ways in which different set theory are used. To do so,IJOART an exploratory research approach has been applied. Keywords: Set theory, Universal Set, Null Set, Finite Set, Infinite Set, Equivalent Set, Venn diagram and De Morgan’s law. Introduction: Set theory is a mathematical abstract concerned with the grouping of sets of numbers that have commonality. For example, all even numbers make up a set, and all odd numbers comprise a set. All numbers that end in zero make up a set of numbers that can be divided by 10. Using and comparing sets enables the creation of theories and rules that have practically unlimited scope, whether in mathematics or applied to areas such as business. Applied to business operations, set theory can assist in planning and operations. Every element of business can be grouped into at least one set such as accounting, management, operations, production and sales. Within those sets are other sets. In operations, for example, there are sets of warehouse operations, sales operations and administrative operations. In some cases, sets intersect -- as sales operations can intersect the operations set and the sales set. IJOART Copyright © 2017 SciResPub. International Journal of Advancements in Research & Technology, Volume 6, Issue 4, April-2017 ISSN 2278-7763 2 Objective of the study: The main objective of the study is to explore different types of set theory and their dynamic usage in the different areas of problems. Methodology of the study: Exploratory research approach has been used to conduct the research program successfully. Analysis of the study and Findings: Definition of set: A set is a collection of well defined objects or elements having certain properties. Each object comprising a set is called element of the set. Usually, a set is denoted by the capital letters A, B, C,D,…… etc and elements are denoted by the small letters a, b, c, d, ……… etc and also the elements of the set are placed within a braces or second bracket. Example: (i) A set of all vowels, A={a, e, i, o, u} (ii) A set of even number from 1 to 11 B={2, 4, 6, 8, 10} IJOART Properties of set: (i) It should be well defined (ii) It is the collection of all objects (iii) Elements of set must be well-distinguished and independent. (iv) Elements of set must be homogeneous. (v) Elements of set must be written second bracket {} and separated by commas(,) Elements of a Set The objects used to form a set are called its element or its members.Generally, the elements of a set are written inside a pair of curly (idle) braces and are represented by commas. The name of the set is always written in capital letter. Examples to find the elements or members of a set: 1. A = {v, w, x, y, z} Here ‘A’ is the name of the set whose elements (members) are v, w, x, y, z. IJOART Copyright © 2017 SciResPub. International Journal of Advancements in Research & Technology, Volume 6, Issue 4, April-2017 ISSN 2278-7763 3 2. If a set A = {3, 6, 9, 10, 13, 18}. State whether the following statements are ‘true’ or ‘false’: (i) 7 ∈ A (ii) 12 ∉ A (iii) 13 ∈ A (iv) 9, 12 ∈ A (v) 12, 14, 15 ∈ A Different Notations in Sets To learn about sets we shall use some accepted notations for the familiar sets of numbers. Some of the different notations used in sets are: ∈ Belongs ∉ to : or Does not belongs | to ∅ Such that n(A Null set or empty set ) Cardinal number of the set A ∪ Union of twoIJOART sets ∩ Intersection of two sets N Set of natural numbers = {1, 2, 3, ……} W Set of whole numbers = {0, 1, 2, 3, ………} I or Set of integers = {………, -2, -1, 0, 1, 2, ………} Set of all positive Z integers Set of Z+ all rational numbers Q Set of all positive rational numbers Q+ Set of all real numbers R Set of all positive real numbers R+ Set of all complex numbers C These are the different notations in sets generally required while solving various types of problems on sets. Different Types of Set: Sets may be of various types. We give below a few of them... IJOART Copyright © 2017 SciResPub. International Journal of Advancements in Research & Technology, Volume 6, Issue 4, April-2017 ISSN 2278-7763 4 Finite Set: When the elements of a set can be counted by finite number of elements then the set is called a finite set. The following are the examples of finite sets: A ={ } B = { } C = { } In all above sets the elements can be counted by a finite number. It should br denoted that a set containing a very large number of elements is also a finite set. Thus, the set of all human beings in Bangladesh, the set of all integers between -1 crore and +1 crore all finite sets. Infinite Set: If the elements of a set cannot be counted in a finite number, the set is called an infinite set. The following are examples of infinite sets: ={ } B = { } C = { } Empty or Null Set: Any set which has no elements in it is called an empty set, or a null set or a void set. The symbol used to denote an empty is a Greek letter (read as phi), i.e., zero with a slash through it.IJOART Here the rule or the property describing a given set is such that no element can be included in the set. The following are a few examples of empty set: (i) The set of people who have travelled from the earth t to the sun is an empty set because none has travelled so far. (ii) A= { } Singleton:: A Set containing only one element is called a singleton or a unit set. For example.. A = {x: x is neither prime nor composite} It is a singleton set containing one element, i.e., 1. • B = {x: x is a whole number, x < 1} this set contains only one element 0 and is a singleton set. Equal Set: Two sets A and B are said to be equal if every element of A is also an element of B, and every element of B is also an element of A , i.e. We find that A=B=C, as each set contains the same elements namely a, c, h, m, r irrespective of their order. Hence the sets are equal. It may be noted that the order of elements or the repetition of elements does not matter in set theory. IJOART Copyright © 2017 SciResPub. International Journal of Advancements in Research & Technology, Volume 6, Issue 4, April-2017 ISSN 2278-7763 5 Let { } { } and { } . This A=B=C since each element which belongs to anyone of the sets also belongs to the other two sets. Hence the sets are equal. Equivalent Sets: If the elements of one set can be put into one to one correspondence with the elements of another set, then the two sets are called equivalent sets. For example.. Here A and C are equal sets while A and B are equivalent sets. Let { } and { } Here the elements of A can be put into one to one correspondence with those of B .Thus a, b, c, d 3, 2, 2, 3 Hence A Subset: If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B and we write it as A ⊆ B or B ⊇ A The symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’ • every set is a subset of itself, i.e., A ⊂ A, B ⊂ B. • Empty set is IJOARTa subset of every set. • Symbol ‘⊆’ is used to denote ‘is a subset of’ or ‘is contained in’. • A ⊆ B means A is a subset of B or A is contained in B. • B ⊆ A means B contains A. For example; 1. Let A = {2, 4, 6} B = {6, 4, 8, 2} Here A is a subset of B Since, all the elements of set A are contained in set B. But B is not the subset of A Since, all the elements of set B are not contained in set A. 2. The set N of natural numbers is a subset of the set Z of integers and we write N ⊂ Z.